$X$ is a topological space, I need to give example such that for $x_1,x_0 \in X$
$\pi_1(X,x_0) \not \cong \pi_1(X,x_1)$
I think the example is somehow related to the fact that $X$ is not path connected so we cant find continuous path between $x_0$ and $x_1$
As suggested in the comments, take any two spaces that do not have the same fundamental group, for instance, an interval $I$ and the cicle $S^1$. Then take the disjoint union $I \cup S^1$ as your space $X$. The fundamental group based at points in $I$ is the zero group, and the fundamental group based at points in $S^1$ is the integers.
Generally, if the space is path connected, then two fundamental groups at different base points are always isomorphic. Thus, any example fitting your question must be of the form that the space has (at least) two path connected components, which have different fundamental groups.